On Additive Divisor Sums and Partial Divisor Functions
aa r X i v : . [ m a t h . N T ] M a r ON ADDITIVE DIVISOR SUMS AND PARTIAL DIVISOR FUNCTIONS
J.C. ANDRADE AND K. SMITH
Abstract.
We establish asymptotic formulae for various correlations involving generaldivisor functions d k ( n ) and partial divisor functions d l ( n, A ) = P q | n : q ≤ n A d l − ( q ), where A ∈ [0 ,
1] is a parameter and k, l ∈ N are fixed. Our results relate the parameter A to thelengths of arithmetic progressions in which d k ( n ) is uniformly distributed. As applicationsto additive divisor sums, we establish new lower bounds and a new equivalent conditionfor the conjectured asymptotic. We also prove a Tauberian theorem for general additivedivisor sums. Contents
1. Introduction 21.1. Additive divisor sums 21.2. Exponents of distribution 41.3. Partial divisor functions 52. Results 72.1. On partial divisor functions 72.2. On additive divisor sums 83. Definitions 94. Proofs 104.1. Theorem 2.3 104.2. Theorems 2.1 and 2.2 194.3. Theorem 2.4 224.4. The coefficients in the case k = l = 2 24References 26 Date : March 6, 2019.2010
Mathematics Subject Classification.
Primary 11N37; Secondary 11M06.
Key words and phrases. divisor function, additive divisor sum, arithmetic progression, uniform distribu-tion, exponent of distribution. Introduction
The focus of this paper is the problem of finding asymptotic formulae for ‘additive divisorsums’. That is, correlations D h,k,l ( x ) = X n ≤ x d k ( n + h ) d l ( n ) , (1.1)where h, k, l ∈ N are fixed and d k ( n ) denotes the number of ordered ways of writing n as aproduct of k factors. In other words, this is the problem of counting the number of orderedsolutions of the Diophantine equation h = n · · · n k − m · · · m l where ( m , ..., m l ) ∈ N l , ( n , ..., n k ) ∈ N k and n · · · n k ≤ x .Our results on the correlations in (1.1) are given in Section 2.2. These results are immediatecorollaries of the results given in Section 2.1, which deal with correlations involving d k ( n )and partial divisor functions d l ( n, A ) = X q | n : q ≤ n A d l − ( q ) A ∈ (0 , . (1.2)As such, we emphasise the results given in Section 2.1.1.1. Additive divisor sums.
When k = l , the correlations in (1.1) arise in connectionwith the problem of finding asymptotic formulae for the 2 k th moments of the Riemannzeta function on the critical line. This connection was first exploited by Ingham [20] inthe course of proving his asymptotic formula for the fourth moment of the Riemann zetafunction. Ingham proved that D h, , ( x ) ∼ π σ − ( h ) log x where σ z ( n ) = P d | n d z , and subsequently Estermann [11] established the asymptotic ex-pansion D h, , ( x ) = xP h, , (log x ) + O (cid:16) x / ǫ (cid:17) (1.3)where P h, , is a polynomial of degree 2. Estermann demonstrated that D h, , ( x ) is relatedto the spectral theory of modular forms - his result made crucial use of a non-trivial boundfor Kloosterman sums. Heath-Brown [16] subsequently used Weil’s improved bound [38] forKloosterman sums to obtain the error term O (cid:0) x / ǫ (cid:1) in (1.3), which was later improvedby Motohashi [28] to O (cid:0) x / ǫ (cid:1) uniformly for h ≤ x / . Each of these improvementslead to corresponding improvements of the error term in the asymptotic expansion for the N ADDITIVE DIVISOR SUMS AND PARTIAL DIVISOR FUNCTIONS 3 fourth moment of the Riemann zeta function.Due to the work of Hooley [19], Linnik [25], Fouvry and Tenenbaum [13], Heath-Brown[17], Drappeau [9], Motohashi [27], Deshouillers and Iwaniec [8], Bykovski and Vinogradov[3] and Topacogullari [34, 35, 36], it is now also known that for any fixed k there is a δ > P h,k, of degree k such that D h,k, ( x ) = xP h,k, (log x ) + O h,k ( x − δ ) . (1.4)Despite these significant advances, asymptotic formulae for D h,k,l ( x ) remain elusive whenboth k, l ≥
3. The central conjecture—formulated by Conrey and Gonek [5] and Ivi´c[21, 22] via the ‘ δ -method’ of Duke, Friedlander and Iwaniec [10], and recently refined byNg and Thom [30] and Tao [31]—is as follows. Conjecture 1.1. If h, k, l ∈ N with k, l fixed and h = O ( x − ǫ ) for each fixed ǫ > , thenthere is a δ > and a polynomial P h,k,l of degree k + l − such that D h,k,l ( x ) = xP h,k,l (log x ) + O ǫ,k,l ( x − δ ) . The asymptotic is conjectured to be D h,k,l ( x ) x log k + l − x ∼ C k,l f k,l ( h )( k − l − as x → ∞ , where C k,l = Y p (cid:0) − p − (cid:1) l − + (cid:0) − p − (cid:1) k − − (cid:0) − p − (cid:1) k + l − (1.6) and f k,l ( h ) = Y p | h (1 − p − ) P γ d l − ( p α ) P ∞ α d k ( p β ) p − β + d k ( p γ ) P ∞ γ +1 d l − ( p α ) p − α (1 − p − ) − k + (1 − p − ) − l − where h = Q p γ . The general form of the coefficients C k,l and f k,l ( h ) appearing in (1.5) were calculated byNg and Thom [30] based on the techniques introduced by Conrey and Gonek [5], and thesame prediction was made by Tao [31] based on pseudorandomness heuristics.The asymptotic order of D h,k,l ( x ) is fairly well understood. Regarding upper bounds, itfollows from the general theorem of Nair and Tenenbaum [29] that D h,k,l ( x ) = O h,k,l ( x log k + l − x ) , (1.8)with uniformity in the h aspect following from the work of Henriot [18]—the paper of Ngand Thom [30] discusses these matters in detail. However, when k, l ≥
3, it is notable that
J.C. ANDRADE AND K. SMITH explicit bounds on the size of the constant implied in (1.8) have not yet appeared in theliterature. Regarding lower bounds, the best general result in the literature is due to Ngand Thom [30], who showed that for k, l ≥ B k,l > h ≤ exp (cid:16) B k,l (log x log log x ) (min( k,l ) − / (min( k,l ) − . (cid:17) we have D h,k,l ( x ) x log k + l − x ≥ (cid:18) O k,l (cid:18) log log h log x (cid:19)(cid:19) − k − l C k,l f k,l ( h )( k − j − . (1.9)Regarding averages over h , Matomaki, Radziwill and Tao [26] have recently shown thatthe conjectured asymptotic (1.5) holds for k , l ≥ h ≤ H , provided that x / ǫ ≤ H ≤ x − ǫ , improving on previous work of Baier, Browning, Marasingha andZhao [1] on the case k = l = 3.1.2. Exponents of distribution.
The problem of the asymptotic behaviour of additivedivisor sums is closely related to the problem of improving the ‘exponent of distribution’for the generalised divisor problem in arithmetic progressions. An exponent of distributionis a lower bound on the lengths of arithmetic progressions n ≡ h (mod q ), ( h, q ) = g , inwhich d k ( n ) is uniformly distributed. Definition 1.1.
A real number < θ g,k ≤ is an exponent of distribution for d k ( n ) if forevery q ≤ x θ g,k − ǫ and each residue class h q ) , we have X n ≤ xn ≡ h (mod q ) d k ( n ) = 1 φ ( q/g ) Res x s s X ( n,q )= g d k ( n ) n s , s = 1 + O ǫ,δ,k (cid:18) x − δ φ ( q/g ) (cid:19) (1.10) for some fixed δ > and ǫ > . An alternative way of writing (1.10) is X n ≤ xn ≡ h (mod q ) d k ( n ) = 1 φ ( q/g ) X n ≤ x/g χ ( n ) d k ( gn ) + O ǫ,δ,k (cid:18) x − δ φ ( q/g ) (cid:19) (1.11)where χ is the principal Dirichlet character to the modulus q/g . Definition 1.1 is moti-vated by the fact that we expect d k ( n ) to be uniformly distributed even in short arithmeticprogressions (i.e. with q ≤ x − ǫ for every fixed ǫ > θ g,k = 1 for all k provided that g is not large, say g ≤ x − ǫ . N ADDITIVE DIVISOR SUMS AND PARTIAL DIVISOR FUNCTIONS 5
Currently however, we can only prove uniform distribution in sufficiently long arithmeticprogressions. The best results in the literature are as follows. For k = 2, Hooley [19] es-tablished that we may take θ , = 2 /
3. We have θ g, = 21 /
41 for all g due to Heath-Brown[17], θ , = 1 / θ , = 9 / θ , = 5 /
12 and θ ,k = 8 / k for k ≥ k >
2, the only known k for which an exponentof distribution greater than 1 / k = 3, and both proofs (including the infe-rior exponent 58 /
115 due to Friedlander and Iwaniec [14]) depend on Deligne’s Riemannhypothesis for algebraic varieties over finite fields. For specific moduli, further incrementshave also been achieved. For instance, via general estimates for sums of trace functionsover finite fields twisted by Fourier coefficients of Eisenstein series, Fouvry, Kowalski andMichel [12] have shown that (1.10) holds for k = 3 for all primes q ≤ x / (albeit with x − δ in the error term replaced with log − C x for every C > k = 3, the above results on exponents ofdistribution are stated only for g = 1. This is usually because g = 1 is the only valuethat is required in applications to primes in arithmetic progressions. Yet, for applicationsto additive divisor sums, we require exponents of distribution for all g in some range as x → ∞ . In this regard, by generalising Heath-Brown’s argument, Chace [4] has shown that θ k = max (cid:18) k , θ ,k + (1 − kθ ,k ) lim sup x →∞ log g log x (cid:19) (1.12)is an exponent of distribution for d k ( n ) for all g .1.3. Partial divisor functions.
A central principle in this paper is that partial divisorfunctions d k ( n, A ) = X d | nd ≤ n A d k − ( d ) A ∈ (0 , d k ( n ) in arithmetic progressions. This property is es-sential in applications to correlation problems such as (1.1). We return to this in due course.The pointwise relationship between d k ( n ) and d k ( n, A ) is generally unpredictable. In thisregard, Tenenbaum [32] showed that lim n →∞ n ∈ S d ( n, A ) d ( n )(1.14)does not exist for any fixed A ∈ (0 ,
1) when S ⊆ N has positive measure. Furthermore,Tenenbaum [33] showed that for every pair A, B ∈ (0 ,
1) there is an S of positive measurein which d ( n, A ) = d ( n, B ) for every n ∈ S . Presumably, the same conclusions hold forevery k ≥ J.C. ANDRADE AND K. SMITH
On the other hand, the limit in (1.14) exists on particular sets S of zero measure. Forexample, if p is prime then we havelim α →∞ d k ( p α , A ) d k ( p α ) = A k − (1.15)and so, by partial summation and (1.15), it follows that X α ≤ X a α d k ( p α , A ) ∼ A k − X α ≤ X a α d k ( p α )(1.16)whenever ( a α ) is a sequence of non-negative real numbers such that P α ≤ X a α → ∞ .On average, the relationship between d k ( n, A ) and d k ( n ) is predictable. In this direction,Deshoulliers, Dress and Tenenbaum [7] proved that the mean value of d ( n, A ) /d ( n ) con-verges to an arcsine distribution. This has been generalised by Bareikis [2], giving a betadistribution 1 x X n ≤ x d k ( n, A ) d k ( n ) ∼ R A u − /k (1 − u ) /k − du Γ(1 /k )Γ(1 − /k )(1.17)uniformly for 0 ≤ A ≤ x → ∞ , for any fixed k ≥ f : N → C , then X n ≤ x X d | nd ≤ n A f ( d ) = x X n ≤ x A f ( n ) n + O X n ≤ x A | f ( n ) | n − /A A ∈ (0 , , (1.18)uniformly for A ≥ A >
0. Taking f ( n ) = d k − ( n ) in (1.18) and using Perron’s formulato evaluate the r.h.s, it is easily seen that d k ( n, A ) approximates A k − d k ( n ) in the mean,that is X n ≤ x d k ( n, A ) = A k − X n ≤ x d k ( n ) + O A (cid:16) x log k − x (cid:17) . (1.19)Here, it is notable that the non-multiplicativity of d k ( n, A ) is crucial to the quality ofthe approximation in (1.19). Indeed, since d k ( p α , A ) = d k ( p ⌊ αA ⌋ ), the mean value of Q p | n d k ( p α , A ) exists for every A < k ∈ N , whereas the mean of d k ( n ) is ∼ log k − x/ ( k − N ADDITIVE DIVISOR SUMS AND PARTIAL DIVISOR FUNCTIONS 7
An elementary refinement of (1.18) is that X n ≤ xn ≡ h (mod q ) X d | nd ≤ n A f ( d ) = xq X n ≤ x A ( n,q ) | h ( n, q ) f ( n ) n + O X n ≤ x A | f ( n ) | n − /A , (1.20)which may be proved by interchanging the order of summation and trivially estimatingthe length of the resulting arithmetic progression. However, in applications to correlationproblems, the error term in (1.20) is not strong enough. Typically we require an additionalfactor of 1 /q , uniformly for q ≤ x C as x → ∞ for some C > − A . Our first theorem(Theorem 2.1) establishes the requisite refinement of (1.20) in the case f ( n ) = d k − ( n ), witha fairly strong value of C . Our second theorem (Theorem 2.2) is deduced from Theorem2.1. 2. Results
On partial divisor functions.
The theorems in this section do all the work in prov-ing the corollaries on additive divisor sums in Section 2.2. Theorems 2.1 and 2.2 are provedin Section 4.2 and Theorem 2.3 is proved in Section 4.1. In light of the structural connec-tion to additive divisor sums, theorems of this type are potentially of further use in suchapplications. Moreover, in accordance with the conjecture that θ k = 1, we expect that theranges of A , B and q for which these formulae hold may be improved significantly.Theorem 2.1 shows that d k ( n, A ) provides a robust approximation to A k − d k ( n ) in arith-metic progressions. Theorem 2.1. If h, k ∈ N are fixed and q ≤ x min( θ k ,Aθ k − ) − ǫ , then X n ≤ xn ≡ h (mod q ) d k ( n, A ) = A k − X n ≤ xn ≡ h (mod q ) d k ( n ) + O A,ǫ,h,k x log k − xq ! . (2.1) In other words X n ≤ xn ≡ h (mod q ) d k ( n, A ) = xq X n ≤ x A ( n,q ) | h ( n, q ) d k − ( n ) n + O A,h,k x log k − xq ! . (2.2)Theorem 2.2 gives an asymptotic formula for the correlation of d k ( n, A ) with d l ( n, B ). Theorem 2.2. If A ≤ , B < min( θ k , Aθ k − ) and h, k, l ∈ N are fixed, then X n ≤ x d k ( n + h, A ) d l ( n, B ) = A k − B l − C k,l f k,l ( h )( k − l − x log k + l − x + O A,B,h,k,l (cid:16) x log k + l − (cid:17) , (2.3) J.C. ANDRADE AND K. SMITH where C k,l and f k,l ( h ) are defined in (1.6) and (1.7). Theorem 2.3 gives an asymptotic expansion with power saving error term for the correlationof d k ( n ) and d l ( n, A ). Theorem 2.3. If A < θ k and h, k, l ∈ N are fixed, then there is a δ > and a polynomial P A,h,k,l of degree k + l − such that X n ≤ x d k ( n + h ) d l ( n, A ) = xP A,h,k,l (log x ) + O A,δ,h,k,l (cid:16) x − δ (cid:17) . (2.4) An explicit formula for P A,h,k,l is given in (4.20). In particular, the coefficient of theleading term is A l − C k,l f k,l ( h ) / ( k − l − . We note that if θ k > / l = 2, then A = 1 / k = l = 2 to reproduce Estermann’s asymptotic expansion(1.3) explicitly.2.2. On additive divisor sums.
Corollaries I, II and III follow immediately from thetheorems of Section 2.1.Corollary I sharpens the lower bound (1.9) given by Ng and Thom in [30] when h is fixedand k is sufficiently large in comparison with l . Corollary I.
For fixed h, k, l ∈ N we have lim inf x →∞ D h,k,l ( x ) x log k + l − x ≥ θ l − k C k,l f k,l ( h )( k − j − . (2.5) Proof.
Note that d l ( n ) ≥ d l ( n, A ), and use Theorem 2.3. (cid:3) For instance, given Heath-Brown’s exponent θ = 21 /
41, it follows from Corollary I thatlim inf x →∞ D h, , ( x ) x log x ≥ . C , f , ( h )4 . Corollary II gives an equivalent condition for the conjectured asymptotic (1.5).
Corollary II.
For fixed h, k, l ∈ N , the asymptotic (1.5) holds if and only if X n ≤ x d k ( n + h ) (cid:16) d l ( n ) − B − l d l ( n, B ) (cid:17) = o (cid:16) x log k + l − x (cid:17) (2.6) for some (and hence every) B < θ k .Proof. Compare (1.5) with Theorem 2.3 or Theorem 2.2 with A = 1. (cid:3) N ADDITIVE DIVISOR SUMS AND PARTIAL DIVISOR FUNCTIONS 9
In support of the plausibility of (2.6), we note that
Corollary III. If A < θ l , B < min( θ k , Aθ k − ) and h, k, l ∈ N are fixed, then X n ≤ x d k ( n + h, A ) (cid:16) d l ( n ) − B − l d l ( n, B ) (cid:17) = O A,B,h,k,l (cid:16) x log k + l − x (cid:17) . (2.7) Proof.
This follows from Theorem 2.2 by swapping variables
A, B and k, l . (cid:3) The last result of this section is Theorem 2.4. This is a Tauberian theorem and may beviewed as an analogue of the relationship between the Prime Number Theorem and thenon-vanishing of ζ (1 + it ), in which we view D h,k,l ( x ) analogously to π ( x ). Theorem 2.4 isproved in Section 4.3. Theorem 2.4.
Let h, k, l ∈ N and ≤ y < ∞ be fixed, then the function D h,k,l ( s, y ) = ∞ X d k ( n + h ) d l (cid:16) n, y log n (cid:17) ( n + h ) s ( σ > has an analytic continuation to the complex plane except for a pole of order k − at s = 1 and, if the limit lim y →∞ D h,k,l (1 + it, y ) is continuous for t = 0 , then we have D h,k,l ( x ) x log k + l − x ∼ C k,l f k,l ( h )( k − l − as x → ∞ . Definitions
Definitions 3.1—3.4 arise in the course of the proofs.
Definition 3.1.
For j ∈ N and s ∈ C , we define ( s − j ζ j ( s ) = ∞ X r =0 a r ( j ) r ! ( s − r and n ! d n ds n ( s − j ζ j ( s ) s (cid:12)(cid:12)(cid:12)(cid:12) s =1 = n X r =0 ( − n − r a r ( j ) r ! = c n ( j ) . Definition 3.2.
For h, k, l ∈ N , h = Q p γ , ℜ w > − − σ − l − and σ > −ℜ w , we also define C k,l ( s, w ) = Y p (1 − p − w − ) l − + (1 − p − s ) k − p − − (1 − p − s ) k (1 − p − w − ) l − − p − , f h,k,l ( s, w ) = Y p | h (1 − p − ) P γ d l − ( p α ) P ∞ α d k ( p β ) p − βs − αw + d k ( p γ ) P ∞ γ +1 d l − ( p α ) p − α ( w +1) (1 − p − )(1 − p − s ) − k + (1 − p − w − ) − l − , and C k,l ( s, w ) f h,k,l ( s, w ) = ∞ X ϕ h,k,l ( q, s ) q w . (3.1) Definition 3.3.
For m < k and n < l we define b h,k,l,m,n = k − − m X i =0 l − − n X j =0 a l − − n − j ( l − c k − − m − i ( k )( l − − n − j )! ∂ i i ! ∂s i ∂ j j ! ∂w j ∞ X ϕ h,k,l ( q, s ) q w (cid:12)(cid:12)(cid:12)(cid:12) w =0 ,s =1 (3.2) and note that the Dirichlet series in (3.2) converge absolutely. In particular, we have b h,k,l,k − ,l − = C k,l f k,l ( h ) where C k,l and f k,l ( h ) are defined in (1.6) and (1.7). Definition 3.4.
Lastly, for m < k + l − and < A ≤ we define a A,h,k,l,m = ( − m k − X j = m − l +2 (cid:18) ij (cid:19) j X r = m − l +2 r − m + l − X v =0 ( − A ) r − j − v + l − a v ( l − v − l + 1) r v ! × (cid:18) l − v − j − r (cid:19) k − X i = j c k − − i ( k ) i ! ∂ i − j ∂s i − j ∂ j + l − r − ∂w j + l − r − ∞ X ϕ h,k,l ( d, s ) d w (cid:12)(cid:12)(cid:12)(cid:12) w =0 ,s =1 . (3.3) 4. Proofs
Theorem 2.3.
Proof.
We have X n ≤ x d k ( n + h ) d l ( n, A ) = X n ≤ x d k ( n + h ) X q | nq ≤ n A d l − ( q )= X q ≤ x A d l − ( q ) X q /A + h ≤ n ≤ x + hn ≡ h (mod q ) d k ( n ) . (4.1)Using Definition 1.1 to evaluate the inner summations on the r.h.s of (4.1), for 0 < A < θ k we see that (4.1) is X n ≤ x d k ( n + h ) d l ( n, A ) = X q ≤ x A d l − ( q ) φ (cid:16) q ( h,q ) (cid:17) Res ( x + h ) s s X ( n,q )=( h,q ) d k ( n ) n s , s = 1 N ADDITIVE DIVISOR SUMS AND PARTIAL DIVISOR FUNCTIONS 11 − X q ≤ x A d l − ( q ) φ (cid:16) q ( h,q ) (cid:17) Res ( q /A + h − δ A ( q )) s s X ( n,q )=( h,q ) d k ( n ) n s , s = 1 (4.2) + O A,δ,k x − δ X q ≤ x A d l − ( q ) φ (cid:16) q ( h,q ) (cid:17) + O A,δ,k x − δ X q ≤ x A d l − ( q )( q /A + h − δ A ( q )) − δ φ (cid:16) q ( h,q ) (cid:17) where δ A ( q ) = 0 or 1 depending on whether q /A is an integer or not. The summations inthe error terms in the third and fourth lines of (4.2) are O A,h (log l − x ), so it remains toevaluate the first two terms.4.1.1. Evaluation of the primary term.
We begin by evaluating the first term on the r.h.s of(4.2). Let χ denote the principal character to the modulus q/g , where q = Q p α , h = Q p γ and g = ( h, q ) = Q p δ so δ = min( α, γ ). We have X ( n,q )= g d k ( n ) n s = ∞ X χ ( n ) d k ( gn )( gn ) s = Y p ∞ X d k ( p β + δ ) χ ( p β ) p − ( β + δ ) s = L k ( s, χ ) b h,k ( s, q )where b h,k ( s, q ) = Y p | g (1 − χ ( p ) p − s ) k ∞ X δ d k ( p β ) χ ( p β − δ ) p − βs is a multiplicative function of g for all k, s . By Cauchy’s theorem, the first term on ther.h.s of (4.2) is= 1( k − ∂ k − ∂s k − ( s − k ζ k ( s ) Z h,k,l (cid:0) s, x A (cid:1) ( x + h ) s s (cid:12)(cid:12)(cid:12)(cid:12) s =1 = 1( k − k − X i =0 (cid:18) k − i (cid:19) ∂ i ∂s i Z h,k,l (cid:0) s, x A (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) s =1 ∂ k − − i ∂s k − − i ( s − k ζ k ( s )( x + h ) s s (cid:12)(cid:12)(cid:12)(cid:12) s =1 (4.3)where Z h,k,l (cid:0) s, x A (cid:1) = X q ≤ x A d l − ( q ) φ (cid:16) q ( h,q ) (cid:17) Y p | q ( h,q ) (cid:0) − p − s (cid:1) k b h,k ( s, q ) , (4.4) and so our first task is to find asymptotic formulae for ∂ i ∂s i Z h,k,l ( s, Q ) at s = 1 as Q → ∞ .To proceed, we note that the factor d l − ( q ) φ (cid:16) q ( q,h ) (cid:17) Y p | q ( q,h ) (cid:0) − p − s (cid:1) k of the summand in (4.4) is a multiplicative function of q for all h, k, s , and we shall nowshow that b h,k ( s, q ) is also. From (4.3) we have b h,k ( s, q ) = Y p | ( q,h ) p | q ( q,h ) d k ( p δ ( q ) ) p δ ( q ) s Y p | ( q,h ) p ∤ q ( q,h ) (1 − p − s ) k ∞ X δ ( q ) d k ( p β ) p − βs . (4.5)If q = rt with ( r, t ) = 1, we have ( rt, h ) = ( r, h )( t, h ) and δ ( rt ) = δ ( r ) + δ ( t ) for every p sothe inclusion p | ( rt, h ) in (4.5) is multiplicative, that is b h,k ( s, rt ) = Y p | ( r,h ) p | r ( r,h ) t ( t,h ) d k ( p δ ( r ) ) p δ ( r ) s Y p | ( r,h ) p ∤ r ( r,h ) t ( t,h ) (1 − p − s ) k ∞ X δ ( r ) d k ( p β ) p − βs × Y p | ( t,h ) p | r ( r,h ) t ( t,h ) d k ( p δ ( t ) ) p δ ( t ) s Y p | ( t,h ) p ∤ r ( r,h ) t ( t,h ) (1 − p − s ) k ∞ X δ ( t ) d k ( p β ) p − βs . (4.6)Since p | r implies p ∤ t , the intersection of the sets p | ( r, h ) and p | t/ ( t, h ) in (4.6) is alreadyempty. It follows that if p | ( r, h ) then the inclusion p | t/ ( t, h ) and exclusion p ∤ t/ ( t, h ) issuperfluous, and vice-versa. Therefore b h,k ( s, rt ) = Y p | ( r,h ) p | r ( r,h ) d k ( p δ ( r ) ) p δ ( r ) s Y p | ( r,h ) p ∤ r ( r,h ) (1 − p − s ) k ∞ X δ ( r ) d k ( p β ) p − βs × Y p | ( t,h ) p | t ( t,h ) d k ( p δ ( t ) ) p δ ( rt ) s Y p | ( t,h ) p ∤ t ( t,h ) (1 − p − s ) k ∞ X δ ( t ) d k ( p β ) p − βs = b h,k ( s, r ) b h,k ( s, t ) . Thus Z h,k,l (cid:0) s, x A (cid:1) = X q ≤ x A φ h,k,l ( s, q ) N ADDITIVE DIVISOR SUMS AND PARTIAL DIVISOR FUNCTIONS 13 where the summand φ h,k,l ( s, q ) = d l − ( q ) φ (cid:16) q ( q,h ) (cid:17) Y p | q ( q,h ) (cid:0) − p − s (cid:1) k Y p | ( q,h ) p | q ( q,h ) d k ( p δ ( q ) ) p δ ( q ) s Y p | ( q,h ) p ∤ q ( q,h ) (1 − p − s ) k ∞ X δ ( q ) d k ( p β ) p − βs is a multiplicative function of q for all h, k, l, s .Since φ h,k,l ( s, q ) is multiplicative, we define the Euler productΦ h,k,l ( s, w ) = Y p ∞ X φ h,k,l ( s, p α ) p − αw for values of w ∈ C for which the r.h.s converges absolutely. If p ∤ h then φ h,k,l ( s, p α ) = φ ,k,l ( s, p α ) which, after some routine algebra, givesΦ h,k,l ( s, w ) = C k,l ( s, w ) f h,k,l ( s, w ) ζ l − ( w + 1) , (4.7)where f h,k,l ( s, w ) is defined in (3.1) and the Euler product C k,l ( s, w ) = Y p (cid:0) − p − w − (cid:1) l − + (1 − p − s ) k (cid:0) − (1 − p − w − ) l − (cid:1) − p − (4.8)converges absolutely for ℜ w > − − σ − l − and σ > −ℜ w (this follows from the fact that thelargest power of p appearing in each factor of the Euler product has real part strictly lessthan − s, w are in this range). Consequently, C k,l ( s, w ) is analytic and bounded oncompact subsets of the half planes ℜ w > − − σ − l − and σ > −ℜ w . It follows that for fixed h, i, k, l the Dirichlet series ∂ i ∂s i C k,l ( s, w ) f h,k,l ( s, w ) = ∞ X ∂ i ∂s i ϕ h,k,l ( q, s ) q w is absolutely convergent and bounded for such values of s, w . Thus, using the relation φ h,k,l ( s, q ) = X d | q ϕ h,k,l ( d, s ) d l − ( q/d ) q/d , (4.9) we have ∂ i ∂s i Z h,k,l ( s, Q ) = ∂ i ∂s i X q ≤ Q X d | q ϕ h,k,l ( d, s ) d l − ( q/d ) q/d = ∂ i ∂s i X d ≤ Q ϕ h,k,l ( d, s ) X q ≤ Q/d d l − ( q ) q = ∂ i ∂s i X d ≤ Q ϕ h,k,l ( d, s ) l − X j =0 a l − − j ( l −
1) log j ( Q/d )( l − − j )! j !(4.10) + 12 πi Z ( ǫ − /l ) ζ l − ( w + 1) ( Q/d ) w dww ! = l − X a l − − j ( l − j !( l − − j )! ∂ i ∂s i X d ≤ Q ϕ h,k,l ( d, s ) log j ( Q/d )+ O Q ǫ − /l X d ≤ Q (cid:12)(cid:12)(cid:12)(cid:12) ∂ i ∂s i ϕ h,k,l ( d, s ) (cid:12)(cid:12)(cid:12)(cid:12) d /l − ǫ where the notation R ( c ) in the fourth line of (4.10) denotes integration along a vertical linefrom c − i ∞ to c + i ∞ . That this integral is O (cid:0) ( Q/d ) ǫ − /l (cid:1) follows from classical resultson the error term in the generalised Dirichlet divisor problem (see Titchmarsh [37], forinstance). Expanding log j ( Q/d ) as a polynomial in log Q , (4.10) is= l − X j =0 a l − − j ( l − j !( l − − j )! j X n =0 (cid:18) jn (cid:19) log n Q ∂ i ∂s i X d ≤ Q ϕ h,k,l ( d, s )( − log d ) j − n + O (cid:16) Q ǫ − /l (cid:17) = l − X n =0 log n Qn ! l − − n X j =0 a l − − n − j ( l − j !( l − − n − j )! ∂ i ∂s i ∂ j ∂w j X d ≤ Q ϕ h,k,l ( d, s ) d w (cid:12)(cid:12)(cid:12)(cid:12) w =0 + O (cid:16) Q ǫ − /l (cid:17) . (4.11)We also have N ADDITIVE DIVISOR SUMS AND PARTIAL DIVISOR FUNCTIONS 15 ∂ k − − i ∂s k − − i ( s − k ζ k ( s )( x + h ) s s (cid:12)(cid:12)(cid:12)(cid:12) s =1 = ( x + h )( k − − i )! k − − i X r =0 a r ( k ) r ! k − − i − r X m =0 ( − k − − i − r − m log m ( x + h ) m != ( x + h )( k − − i )! k − − i X m =0 log k − − i − m ( x + h )( k − − i − m )! c m ( k )(4.12)Setting Q = x A in (4.11) and using (4.12), we conlude that (4.3) is= ( x + h ) k − X m =0 l − X n =0 A n b h,k,l,m,n m ! n ! log m ( x + h ) log n x + O ( x + h ) x ǫ − A/l k − X i =0 k − X m = i log k − − m ( x + h ) | c m − i ( k ) | ( k − − m )! X d ≤ x A (cid:12)(cid:12)(cid:12)(cid:12) ∂ i ∂s i ϕ h,k,l ( d, s ) (cid:12)(cid:12)(cid:12)(cid:12) s =1 d /l − ǫ , = x k − X m =0 l − X n =0 A n b h,k,l,m,n m ! n ! log m + n x + O h,k,l (cid:16) x − A/l + ǫ (cid:17) , (4.13)where the coefficients b h,k,l,m,n are defined in Section 3.4.1.2. Evaluation of the secondary term.
We now evaluate the second term on the r.h.s of(4.2). By Cauchy’s theorem, this is= 1( k − ∂ k − ∂s k − ( s − k ζ k ( s ) W h,k,l (cid:0) s, x A (cid:1) s (cid:12)(cid:12)(cid:12)(cid:12) s =1 = 1( k − k − X i =0 (cid:18) k − i (cid:19) ∂ i ∂s i W h,k,l (cid:0) s, x A (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) s =1 ∂ k − − i ∂s k − − i ( s − k ζ k ( s ) s (cid:12)(cid:12)(cid:12)(cid:12) s =1 = k − X i =0 i ! ∂ i ∂s i W h,k,l (cid:0) s, x A (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) s =1 c k − − i ( k ) , (4.14)where W A,h,k,l ( s, Q ) = X q ≤ Q φ h,k,l ( s, q )( q /A + h − δ A ( q )) s . By (4.9) we have ∂ i ∂s i W A,h,k,l ( s, Q ) (cid:12)(cid:12)(cid:12)(cid:12) s =1 = ∂ i ∂s i X d ≤ Q ϕ h,k,l ( d, s ) X q ≤ Q/d d l − ( q )(( qd ) /A + h − δ A ( q )) s q (cid:12)(cid:12)(cid:12)(cid:12) s =1 = i X j =0 (cid:18) ij (cid:19) X d ≤ Q ∂ i − j ∂s i − j ϕ h,k,l ( d, s ) ∂ j ∂s j V A,h,l,Q ( d, s ) (cid:12)(cid:12)(cid:12)(cid:12) s =1 , (4.15)where V A,h,l,Q ( d, s ) = X q ≤ Q/d d l − ( q )(( qd ) /A + h − δ A ( q )) s q and ∂ j ∂s j V A,h,l,Q ( d, s ) (cid:12)(cid:12)(cid:12)(cid:12) s =1 = ( − j X q ≤ Q/d d l − ( q )(( qd ) /A + h − δ A ( q )) log j (( qd ) /A + h − δ A ( q )) q = ( − A ) − j d /A X q ≤ Q/d d l − ( q ) log j ( qd ) q − /A + O A,h,j,l (( Q/d ) ǫ )= ( − A ) − j d /A j X (cid:18) jm (cid:19) log j − m d X q ≤ Q/d d l − ( q ) log m qq − /A + O A,h,j,l (( Q/d ) ǫ ) . (4.16)The inner summation on the r.h.s of (4.16) may be written as X q ≤ Q/d d l − ( q ) log m qq − /A = ( − m πi Z ( ǫ ) d m dw m ζ l − ( w + 1) ( Q/d ) w +1 /A dww + 1 /A , which may be evaluated using Cauchy’s Theorem and classical results on the error termin the generalised Dirichlet divisor problem (see Titchmarsh [37]). The error term is O A,h,l (cid:0) ( Q/d ) /A − /l + ǫ (cid:1) , and the residue at the pole at w = 0 is= ( − m ( m + l − m + l − X v =0 (cid:18) m + l − v (cid:19) ∂ v ∂w v w m + l − ∂ m ∂w m ζ l − ( w + 1) (cid:12)(cid:12)(cid:12)(cid:12) w =0 ∂ m + l − − v ∂w m + l − − v ( Q/d ) w +1 /A w + 1 /A (cid:12)(cid:12)(cid:12)(cid:12) w =0 = ( − m − ( Q/d ) /A l + m − X v =0 a v ( l − v − l + 1) m v ! l + m − − v X r =0 ( − A ) l + m − − v − r log r ( Q/d ) r != ( − m − ( Q/d ) /A l + m − X r =0 log r ( Q/d ) r ! l + m − − r X v =0 ( − A ) l + m − − v − r a v ( l − v − l + 1) m v ! . N ADDITIVE DIVISOR SUMS AND PARTIAL DIVISOR FUNCTIONS 17
As such, (4.16) is= ( − A ) − j Q /A j X (cid:18) jm (cid:19) ( − m − log j − m d l + m − X r =0 log r ( Q/d ) r ! × l + m − − r X v =0 ( − A ) l + m − − v − r a v ( l − v − l + 1) m v !(4.17) + O A,h,j,l (cid:16) Q /A − /l + ǫ d /l − ǫ (cid:17) and so, expanding log j ( Q/d ) as a polynomial in log Q on the r.h.s of (4.17), it follows that(4.15) is= − Q /A i X j =0 (cid:18) ij (cid:19) ( − j j X m =0 (cid:18) jm (cid:19) l + m − X r =0 r X u =0 log u Qu !( r − u )! l + m − − r X v =0 × ( − A ) l + m − − j − v − r a v ( l − v − l + 1) m v ! ∂ i − j ∂s i − j X d ≤ Q ϕ h,k,l ( d, s )( − log d ) j − m + r − u (cid:12)(cid:12)(cid:12)(cid:12) s =1 + O (cid:16) Q /A + ǫ − /l (cid:17) = − Q /A i X u =2 − l log l + u − Q ( l + u − i X j = u (cid:18) ij (cid:19) ( − j j X m = u (cid:18) jm (cid:19) m − u X r =0 r X v =0 × ( − A ) r − j − v +1 a v ( l − v − l + 1) m ( m − r − u )! v ! ∂ i − j ∂s i − j ∂ j + l − r − u − ∂w j + l − r − u − X d ≤ Q ϕ h,k,l ( d, s ) d w (cid:12)(cid:12)(cid:12)(cid:12) s =1 ,v =0 + O (cid:16) Q /A + ǫ − /l (cid:17) = − Q /A i X u =2 − l log l + u − Q ( l + u − i X j = u (cid:18) ij (cid:19) ( − j j X m = u (cid:18) jm (cid:19) m X r = u r − u X v =0 × ( − A ) r − j − u − v +1 a v ( l − v − l + 1) m ( m − r )! v ! ∂ i − j ∂s i − j ∂ j + l − r − ∂w j + l − r − X d ≤ Q ϕ h,k,l ( d, s ) d w (cid:12)(cid:12)(cid:12)(cid:12) s =1 ,v =0 + O (cid:16) Q /A + ǫ − /l (cid:17) = − Q /A i X u =2 − l log l + u − Q ( l + u − i X j = u (cid:18) ij (cid:19) ( − j j X r = u r − u X v =0 ( − A ) r − j − u − v +1 a v ( l − v ! × j X m = r (cid:18) jm (cid:19) ( v − l + 1) m ( m − r )! ! ∂ i − j ∂s i − j ∂ j + l − r − ∂w j + l − r − X d ≤ Q ϕ h,k,l ( d, s ) d w (cid:12)(cid:12)(cid:12)(cid:12) s =1 ,v =0 + O (cid:16) Q /A + ǫ − /l (cid:17) = − Q /A i X u =2 − l log l + u − Q ( l + u − i X j = u (cid:18) ij (cid:19) j X r = u r − u X v =0 ( − A ) r − j − u − v +1 a v ( l − v − l + 1) r v ! × (cid:18) l − v − j − r (cid:19) ∂ i − j ∂s i − j ∂ j + l − r − ∂w j + l − r − X d ≤ Q ϕ h,k,l ( d, s ) d w (cid:12)(cid:12)(cid:12)(cid:12) s =1 ,v =0 + O (cid:16) Q /A + ǫ − /l (cid:17) so (4.14) is= − Q /A k − X u =2 − l log l + u − Q ( l + u − k − X j = u (cid:18) ij (cid:19) j X r = u r − u X v =0 ( − A ) r − j − u − v +1 a v ( l − v − l + 1) r v ! × (cid:18) l − v − j − r (cid:19) k − X i = j c k − − i ( k ) i ! ∂ i − j ∂s i − j ∂ j + l − r − ∂w j + l − r − X d ≤ Q ϕ h,k,l ( d, s ) d w (cid:12)(cid:12)(cid:12)(cid:12) s =1 ,v =0 + O (cid:16) Q /A + ǫ − /l (cid:17) = − Q /A k + l − X u =0 log u Qu ! k − X j = u − l +2 (cid:18) ij (cid:19) j X r = u − l +2 r − u + l − X v =0 ( − A ) r − j − u − v + l − a v ( l − v − l + 1) r v ! × (cid:18) l − v − j − r (cid:19) k − X i = j c k − − i ( k ) i ! ∂ i − j ∂s i − j ∂ j + l − r − ∂w j + l − r − X d ≤ Q ϕ h,k,l ( d, s ) d w (cid:12)(cid:12)(cid:12)(cid:12) s =1 ,v =0 + O (cid:16) Q /A + ǫ − /l (cid:17) . (4.18)Taking Q = x A in (4.18) yields1( k − ∂ k − ∂s k − ( s − k ζ k ( s ) W h,k,l (cid:0) s, x A (cid:1) s (cid:12)(cid:12)(cid:12)(cid:12) s =1 = − x k + l − X u =0 a A,h,k,l,u log u xu ! + O (cid:16) x ǫ − A/l (cid:17) . (4.19)From (4.2), (4.13) and (4.19), for A < θ k we have X n ≤ x d k ( n + h ) d l ( n, A ) = x k − X m =0 l − X n =0 A n b h,k,l,m,n m ! n ! log m + n x + x k + l − X m =0 a A,h,k,l,m log m xm !+ O A,h,k,l (cid:16) x ǫ − A/l (cid:17) + O A,δ,k (cid:16) x − δ (cid:17) , (4.20) N ADDITIVE DIVISOR SUMS AND PARTIAL DIVISOR FUNCTIONS 19 where the coefficients b h,k,l,m,n and a A,h,k,l,u are defined in Section 3. This concludes theproof of Theorem 2.3. (cid:3)
Theorems 2.1 and 2.2.
Proof of Theorem 2.1.
We begin by writing d k ( n, A ) = d k ( n ) − X d | nd
1. This is= 1 φ ( q/ ( h, q )) X n ≤ x A χ ( n ) d k − (( h, q ) n ) X x − A ≤ d ≤ x/n ( h,q ) χ ( d ) , = 1 φ ( q/ ( h, q )) X n ≤ x A χ ( n ) d k − (( h, q ) n ) X d ≤ x/n ( h,q ) χ ( d ) + O A,h,k x log k − xq ! = xq X n ≤ x A χ ( n ) d k − (( h, q ) n ) n + O A,h,k x log k − xq ! . (4.32)Since ( n/ ( h, q ) , q/ ( h, q )) = 1 is equivalent to ( n, q ) = ( h, q ) which implies that ( n, q ) | h , weconclude that (4.22) is xq X n ≤ x A ( n,q ) | h ( n, q ) d k − ( n ) n + O A,h,k x log k − xq ! . (4.33)By partial summation or otherwise, the remainder of the proof is trivial. (cid:3) Proof of Theorem 2.2.
This is a straightforward consequence of Theorem 2.1 and the methodof proof of Theorem 2.3. We have X n ≤ x d k ( n + h, A ) d l ( n, B ) = X q ≤ x B d l − ( q ) X n ≡ h (mod q ) q /B ≤ n ≤ x + h d k ( n, A )= A k − X q ≤ x B d l − ( q ) X n ≡ h (mod q ) q /B ≤ n ≤ x + h d k ( n )+ O A,B,h,k x log k − x X q ≤ x B d l − ( q ) q (4.34)provided that B < min( θ k , Aθ k − ), by Theorem 2.1. The first term on the r.h.s. of (4.34)is identical to (4.1), and the summation in the error term is O (log l − x ). (cid:3) Theorem 2.4.
Proof.
We begin by establishing the analytic continuation of D h,k,l ( s, Q ). We have N ADDITIVE DIVISOR SUMS AND PARTIAL DIVISOR FUNCTIONS 23 D h,k,l ( s, Q ) = ∞ X d k ( n + h ) d l (cid:16) n, Q log n (cid:17) ( n + h ) s = ∞ X d k ( n + h )( n + h ) s X q ≤ Qq | n d l − ( q )= X q ≤ Q d l − ( q ) X n ≡ h (mod q ) n>h d k ( n ) n s = X q ≤ Q d l − ( q ) X n ≡ h (mod q ) d k ( n ) n s − d k ( h ) h s X q ≤ Q d l − ( q ) . We have X n ≡ h (mod q ) d k ( n ) n s = 1 φ (cid:16) qg (cid:17) X χ (cid:16) mod qg (cid:17) χ (cid:18) hg (cid:19) ∞ X χ ( n ) d k ( gn )( gn ) s where ∞ X χ ( n ) d k ( gn )( gn ) s = Y p ∞ X d k ( p β + δ ) χ ( p β ) p − ( β + δ ) s = L k ( s, χ ) b k ( s, χ, g )(4.35)is a meromorphic function of s for all h, k, l .This shows that D h,k,l ( s, Q ) is a meromorphic function of s for all h, k, l, Q , and we observethat D h,k,l ( s, Q ) = ζ k ( s ) Z h,k,l ( s, Q ) + B h,k,l ( s, Q )(4.36)say, where Z h,k,l ( s, Q ) is defined in (4.4) and B h,k,l ( s, Q ) is an analytic function of s for allfixed h, k, l, Q . Writing D h,k,l ( x, Q ) = X n ≤ x d k ( n + h ) d l (cid:18) n, Q log n (cid:19) , we have D h,k,l ( s, Q ) = s Z ∞ D h,k,l ( x, Q ) dx ( x + h ) s +1 and, by (4.36), we have D h,k,l ( s, Q ) = Z h,k,l ( s, Q )( s − k + C h,k,l ( s, Q )(4.37)for σ >
1, where C h,k,l ( s, Q ) = O h,k,l,Q (( s − − k ) as s →
1. By (4.36) we know that D h,k,l (1 + it, Q ) is continuous for t = 0 (in fact it is analytic in a neighbourhood of theline). As such, the Delange-Ikehara Tauberian theorem [6] applies, i.e.lim x →∞ D h,k,l ( x, Q ) x log k − x = Z h,k,l (1 , Q ) . Arguing in the same way as in the proof of Theorem 2.3, we have Z h,k,l (1 , Q )log l − Q = C k,l f k,l ( h )( k − l − O h,k,l (cid:18) Q (cid:19) . Now, if lim Q →∞ D h,k,l (1 + it, Q ) is continuous when t = 0, then the restriction that Q isfixed in (4.37) can be removed and the Delange-Ikehara Tauberian theorem still applies.Taking Q = x we have D h,k,l ( x, Q ) = D h,k,l ( x ), which completes the proof. (cid:3) The coefficients in the case k = l = 2 . We conclude this paper with a demonstra-tion that Theorem 2.3 recovers Estermann’s asymptotic expansion for D h, , ( x ) precisely.We take k = l = 2 in (4.20), so that X n ≤ x d ( n + h ) d ( n, A ) = x X m =0 1 X n =0 A n b h, , ,m,n m ! n ! log m + n x + x X m =0 a A,h, , ,m log m xm ! + O A,h (cid:16) x − δ (cid:17) = Ab h, , , , x log x + ( b h, , , , + Ab h, , , , + a A,h, , , ) x log x + ( b h, , , , + a A,h, , , ) x + O A,h (cid:16) x − δ (cid:17) . (4.38)Thus, putting A = 1 / n about n / in (4.38),we obtain D h, , ( x ) = b h, , , , x log x + (cid:0) b h, , , , + b h, , , , + 2 a / ,h, , , (cid:1) x log x + 2 (cid:0) b h, , , , + a / ,h, , , (cid:1) x + O h (cid:16) x − δ (cid:17) . (4.39)We now use Definitions 3.3 and 3.4 to calculate the coefficients in (4.39). We use Ester-mann’s [11] notation σ ′− ( h ) = X d | h log dd , σ ′′− ( h ) = X d | h log dd (4.40)and a ′ = − ∞ X µ ( n ) log nn , a ′′ = ∞ X µ ( n ) log nn . (4.41) N ADDITIVE DIVISOR SUMS AND PARTIAL DIVISOR FUNCTIONS 25
Firstly, for the coefficient of x log x we have b h, , , , = C , (1 , f h, , (1 ,
0) = 6 π σ − ( h ) . Secondly, for the coefficient of x log x , we have2 b h, , , , + b h, , , , + 2 a / ,h, , , = 2 a (1) c (2) C , (1 , f h, , (1 ,
0) + 2 a (1) c (2) ∂∂w C , (1 , w ) f h, , (1 , w ) (cid:12)(cid:12)(cid:12)(cid:12) w =0 + a (1) c (2) C , (1 , f h, , (1 ,
0) + a (1) c (2) ∂∂s C , ( s, f h, , ( s, (cid:12)(cid:12)(cid:12)(cid:12) s =0 − a (1) c (2) C , (1 , f h, , (1 , γC , (1 , f h, , (1 ,
0) + 2 ∂∂w C , (1 , w ) f h, , (1 , w ) (cid:12)(cid:12)(cid:12)(cid:12) w =0 + (2 γ − C , (1 , f h, , (1 ,
0) + ∂∂s C , ( s, f h, , ( s, (cid:12)(cid:12)(cid:12)(cid:12) s =0 − C , (1 , f h, , (1 , π (2 γ − σ − ( h ) + 2 ∂∂w C , (1 , w ) f h, , (1 , w ) (cid:12)(cid:12)(cid:12)(cid:12) w =0 + ∂∂s C , ( s, f h, , ( s, (cid:12)(cid:12)(cid:12)(cid:12) s =0 = (cid:18) π (2 γ −
1) + 2 ∂∂w C , (1 , w ) (cid:12)(cid:12)(cid:12)(cid:12) w =0 + ∂∂s C , ( s, (cid:12)(cid:12)(cid:12)(cid:12) s =0 (cid:19) σ − ( h )+ 6 π (cid:18) ∂∂w f h, , (1 , w ) (cid:12)(cid:12)(cid:12)(cid:12) w =0 + ∂∂s f h, , ( s, (cid:12)(cid:12)(cid:12)(cid:12) s =0 (cid:19) = (cid:18) π (2 γ −
1) + 4 a ′ (cid:19) σ − ( h ) − π σ ′− ( h ) . Lastly, for the coefficient of x we have2 b h, , , , + 2 a / ,h, , , = 2 a (1) c (2) C , (1 , f h, , (1 ,
0) + 2 a (1) C , (1 , ∂∂s f h, , ( s, (cid:12)(cid:12)(cid:12)(cid:12) s =1 + 2 a (1) f h, , (1 , ∂∂s C , ( s, (cid:12)(cid:12)(cid:12)(cid:12) s =1 + 2 a (1) c (2) C , (1 , ∂∂w f h, , (1 , w ) (cid:12)(cid:12)(cid:12)(cid:12) w =0 + 2 c (2) f h, , (1 , ∂∂w C , (1 , w ) (cid:12)(cid:12)(cid:12)(cid:12) w =0 + 2 C , (1 , ∂∂s ∂∂w f h, , ( s, w ) (cid:12)(cid:12)(cid:12)(cid:12) w =0 ,s =1 + 2 ∂∂w C , ( s, w ) (cid:12)(cid:12)(cid:12)(cid:12) w =0 ∂∂s f h, , ( s, w ) (cid:12)(cid:12)(cid:12)(cid:12) s =1 + 2 ∂∂s C , ( s, w ) (cid:12)(cid:12)(cid:12)(cid:12) s =1 ∂∂w f h, , ( s, w ) (cid:12)(cid:12)(cid:12)(cid:12) w =06 J.C. ANDRADE AND K. SMITH + 2 f h, , (1 , ∂∂w ∂∂s C , ( s, w ) (cid:12)(cid:12)(cid:12)(cid:12) w =0 ,s =1 − c (2) C , (1 , f h, , (1 , − f h, , (1 , ∂∂s C , ( s, (cid:12)(cid:12)(cid:12)(cid:12) s =1 − C , (1 , ∂∂s f h, , ( s, (cid:12)(cid:12)(cid:12)(cid:12) s =1 + C , (1 , f h, , (1 , γπ (2 γ − σ − ( h ) − γπ σ ′− ( h ) + 4 γa ′ σ − ( h ) − π (2 γ − σ ′− ( h ) + 2(2 γ − a ′ σ − ( h )+ 24 π σ ′′− ( h ) − a ′ σ ′− ( h ) + 4 a ′′ σ − ( h ) − π (2 γ − σ − ( h ) + 12 π σ ′− ( h ) − a ′ σ − ( h ) + 6 π σ − ( h )= (cid:18) π (2 γ − + 6 π + 4 a ′ (2 γ −
1) + 4 a ′′ (cid:19) σ − ( h ) − (cid:18) π (2 γ −
1) + 8 a ′ (cid:19) σ ′− ( h ) + 24 π σ ′′− ( h ) . Acknowledgment:
We would like to thank Professor Aleksandar Ivi´c, Professor SteveGonek, Professor Zeev Rudnick and Professor Terence Tao for their suggestions on a previ-ous version of this paper. The first author is also grateful to the Leverhulme Trust (RPG-2017-320) for the support through the research project grant “Moments of L -functions inFunction Fields and Random Matrix Theory”. The second author is grateful for a PhDstudentship supported by the College of Engineering, Mathematics and Physical Sciencesat the University of Exeter. References [1] S. Baier, T. D. Browning, G. Marasingha, L. Zhao,
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Department of Mathematics, University of Exeter, Exeter, EX4 4QF, UK
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